
By assumption REF  and corollary 1.1.10 in [1]} \({U^{\prime }_F(tx)}/{U^{\prime }_F}(t) \rightarrow x^{\gamma -1}\)  as \(t\rightarrow \infty \)  locally uniformly in \((0, \infty )\) . Hence
\(\dfrac{\frac{N}{k+s}U^{\prime }_F(N/(k+s))}{\frac{N}{k}U^{\prime }_F(N/k)} = \dfrac{k}{k+s} \dfrac{U^{\prime }_F\left(\frac{N}{k}\frac{k}{k+s}\right)}{U^{\prime }_F\left( \frac{N}{k}\right)} \rightarrow 1.\) 
